كتاب الجبر - W-* Algebras

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كتاب الجبر - W-* Algebras 4a56ed9463ab3dbeccd123a34c0fb78a

The present set of notes treats the general theory of von Neumann
algebras.
I have confined the discussion to algebras acting in separable
Hilbert spaces and have systematically emphasized the use of direct
integral decompositions.
The integral decomposition theory reduces the
study of W-* algebras to the study of factors.
In developing the von
Neumann-Murray analysis of factors I have used the straightforward and
relatively pedestrian methods of the original papers rather than the
elegant but somewhat more abstract approach which one finds in the
well-known work of Dixmier on this subject.
The main remaining problem
in the field is undoubtedly that of the structural classification of
factors.
Proofs of the existence of at least three non-isomorphic
factors of types II and III are given in the present notes.
The results
presented have recently been extended very considerably by Powers in the
type III case; he has succeeded in showing that there are uncountably
many distinct factors of type III.
Strikingly, these examples have been
suggested by the quantum theory of fields, thereby verifying von
Neumann's earlier feeling that W-* algebras would exhibit interesting
relationships to quantum theory.

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